Effects of randomization on asymptotic periodicity of nonsingular transformations

It is known that the Perron--Frobenius operators of piecewise expanding $\mathcal{C}^2$ transformations possess an asymptotic periodicity of densities. On the other hand, external noise or measurement errors are unavoidable in practical systems; therefore, all realistic mathematical models should be regarded as random iterations of transformations. This paper aims to discuss the effects of randomization on the asymptotic periodicity of densities.Comment: 20 pages, 10 figure

Crystal Structure of Cas9 in Complex with Guide RNA and Target DNA

The CRISPR-associated endonuclease Cas9 can be targeted to specific genomic loci by single guide RNAs (sgRNAs). Here, we report the crystal structure of Streptococcus pyogenes Cas9 in complex with sgRNA and its target DNA at 2.5 Å resolution. The structure revealed a bilobed architecture composed of target recognition and nuclease lobes, accommodating the sgRNA:DNA heteroduplex in a positively charged groove at their interface. Whereas the recognition lobe is essential for binding sgRNA and DNA, the nuclease lobe contains the HNH and RuvC nuclease domains, which are properly positioned for cleavage of the complementary and noncomplementary strands of the target DNA, respectively. The nuclease lobe also contains a carboxyl-terminal domain responsible for the interaction with the protospacer adjacent motif (PAM). This high-resolution structure and accompanying functional analyses have revealed the molecular mechanism of RNA-guided DNA targeting by Cas9, thus paving the way for the rational design of new, versatile genome-editing technologies.National Institutes of Health (U.S.) (Grant 5DP1-MH100706

Structural and functional insights into IZUMO1 recognition by JUNO in mammalian fertilization

Kato, K., Satouh, Y., Nishimasu, H. et al. Structural and functional insights into IZUMO1 recognition by JUNO in mammalian fertilization. Nat Commun 7, 12198 (2016). https://doi.org/10.1038/ncomms1219

Crystal Structure and Activity of the Endoribonuclease Domain of the piRNA Pathway Factor Maelstrom

SummaryPIWI-interacting RNAs (piRNAs) protect the genome from transposons in animal gonads. Maelstrom (Mael) is an evolutionarily conserved protein, composed of a high-mobility group (HMG) domain and a MAEL domain, and is essential for piRNA-mediated transcriptional transposon silencing in various species, such as Drosophila and mice. However, its structure and biochemical function have remained elusive. Here, we report the crystal structure of the MAEL domain from Drosophila melanogaster Mael, at 1.6 Å resolution. The structure reveals that the MAEL domain has an RNase H-like fold but lacks canonical catalytic residues conserved among RNase H-like superfamily nucleases. Our biochemical analyses reveal that the MAEL domain exhibits single-stranded RNA (ssRNA)-specific endonuclease activity. Our cell-based analyses further indicate that ssRNA cleavage activity appears dispensable for piRNA-mediated transcriptional transposon silencing in Drosophila. Our findings provide clues toward understanding the multiple roles of Mael in the piRNA pathway

Crystal Structure of Staphylococcus aureus Cas9

Summary The RNA-guided DNA endonuclease Cas9 cleaves double-stranded DNA targets with a protospacer adjacent mot if (PAM) and complementarity to the guide RNA. Recently, we harnessed Staphylococcus aureus Cas9 (SaCas9), which is significantly smaller than Streptococcus pyogenes Cas9 (SpCas9), to facilitate efficient in vivo genome editing. Here, we report the crystal structures of SaCas9 in complex with a single guide RNA (sgRNA) and its double-stranded DNA targets, containing the 5′-TTGAAT-3′ PAM and the 5′-TTGGGT-3′ PAM, at 2.6 and 2.7 Å resolutions, respectively. The structures revealed the mechanism of the relaxed recognition of the 5′-NNGRRT-3′ PAM by SaCas9. A structural comparison of SaCas9 with SpCas9 highlighted both structural conservation and divergence, explaining their distinct PAM specificities and orthologous sgRNA recognition. Finally, we applied the structural information about this minimal Cas9 to rationally design compact transcriptional activators and inducible nucleases, to further expand the CRISPR-Cas9 genome editing toolbox.National Institute of General Medical Sciences (U.S.) (Grant T32GM007753)National Institutes of Health (U.S.) (Award 1DP1-MH100706

Crystal Structure of the Minimal Cas9 from Campylobacter jejuni Reveals the Molecular Diversity in the CRISPR-Cas9 Systems

The RNA-guided endonuclease Cas9 generates a double-strand break at DNA target sites complementary to the guide RNA and has been harnessed for the development of a variety of new technologies, such as genome editing. Here, we report the crystal structures of Campylobacter jejuni Cas9 (CjCas9), one of the smallest Cas9 orthologs, in complex with an sgRNA and its target DNA. The structures provided insights into a minimal Cas9 scaffold and revealed the remarkable mechanistic diversity of the CRISPR-Cas9 systems. The CjCas9 guide RNA contains a triple-helix structure, which is distinct from known RNA triple helices, thereby expanding the natural repertoire of RNA triple helices. Furthermore, unlike the other Cas9 orthologs, CjCas9 contacts the nucleotide sequences in both the target and non-target DNA strands and recognizes the 5′-NNNVRYM-3′ as the protospacer-adjacent motif. Collectively, these findings improve our mechanistic understanding of the CRISPR-Cas9 systems and may facilitate Cas9 engineering. Keywords: CRISPR-Cas system; Cas9; protospacer adjacent motif; RNA triplex; crystal structureUnited States. Department of Energy (Grant DE-FG02-97ER25308)National Institute of Mental Health (U.S.) (Grant 5DP1-MH100706)National Institute of Mental Health (U.S.) (Grant 1R01-MH110049

Structure and Engineering of Francisella novicida Cas9

Summary The RNA-guided endonuclease Cas9 cleaves double-stranded DNA targets complementary to the guide RNA and has been applied to programmable genome editing. Cas9-mediated cleavage requires a protospacer adjacent motif (PAM) juxtaposed with the DNA target sequence, thus constricting the range of targetable sites. Here, we report the 1.7 Å resolution crystal structures of Cas9 from Francisella novicida (FnCas9), one of the largest Cas9 orthologs, in complex with a guide RNA and its PAM-containing DNA targets. A structural comparison of FnCas9 with other Cas9 orthologs revealed striking conserved and divergent features among distantly related CRISPR-Cas9 systems. We found that FnCas9 recognizes the 5′-NGG-3′ PAM, and used the structural information to create a variant that can recognize the more relaxed 5′-YG-3′ PAM. Furthermore, we demonstrated that the FnCas9-ribonucleoprotein complex can be microinjected into mouse zygotes to edit endogenous sites with the 5′-YG-3′ PAM, thus expanding the target space of the CRISPR-Cas9 toolbox